Page 1
Model Predictive Control Approach Based Load Frequency Controller
ALI MOHAMED YOUSEF
Electric Engineering Department,
Faculty of Engineering, Assiut University,
ASSIUT- EGYPT
[email protected]
Abstract:-The present paper investigates the design of Load-Frequency Control ( LFC) system for
improving power system dynamic performance over a wide range of operating conditions based on model
predtictive control MPC technique. The objectives of load frequency control (LFC) are to minimize the
transient deviations in area frequency and tie-line power interchange variables . Also steady state error of the
above variaables forced to be zeros. The two control schems namely Fuzzy logic control and proposed model
predictive control are designed. Both the two controllers empoly the local frequency deviation signal as input
signal. The dynamic model of two-area power system under study is estabilished . To validate the effectiveness
of the proposed MPC controller, two-area power system is simulated over a wide range of operating conditions.
Further, comparative studies between the fuzzy logic controller (FLC), and the proposed MPC load frequency
control are evaluated.
Keywords:- Model predictive control - Fuzzy logic controller, Load Frequency Control, Two area power
system.
1 Introduction
The control strategy is classified into two
controls, firstly, conventional control as integral and
PID control. Secondary is advanced control such as
model predictive control, fuzzy logic control and
neural network and etc., The salient feature of the
fuzzy logic approach is that they provide a model-
free description of control systems and do not require
model identification. The fuzzy LFC systems have
large over and/or under shoots response and large
settling time in non-linear model [1,2]. Although the
active power and reactive power have combined
effects on the frequency and voltage, the control
problem of the frequency and voltage can be
decoupled. The frequency is highly dependent on the
active power while the voltage is highly dependent
on the reactive power. Thus the control issue in
power systems can be decoupled into two
independent problems. One is about the active power
and frequency control while the other is about the
reactive power and voltage control. The active power
and frequency control is referred to as load frequency
control (LFC) [3].
The foremost task of LFC is to keep the frequency
constant against the randomly varying active power
loads, which are also referred to as unknown external
disturbance. Another task of the LFC is to regulate
the tie-line power exchange error [ 4 ].
Therefore, the requirement of the LFC is to be robust
against the uncertainties of the system model and the
variations of system parameters in reality. In
summary, the LFC has two major ssignments, which
are to maintain the standard value of frequency and
to keep the tie-line power exchange under schedule
in the presences of any load changes [3,4]. In
addition, the LFC has to be robust against unknown
external disturbances and system model and
parameter uncertainties. The high-order
interconnected power system could also increase the
complexity of the controller design of the LFC. The
foremost task of LFC is to keep the frequency
constant against the randomly varying active power
loads, which are also referred to as unknown external
disturbance. Another task of the LFC is to regulate
the tie-line power exchange error. A typical large-
scale power system is composed of several areas of
generating units. In order to enhance the fault
tolerance of the entire power system, these
generating units are connected via tie-lines. The
usage of tie-line power imports a new error into the
control problem, i.e., tie-line power exchange error.
When a sudden active power load change occurs to
an area, the area will obtain energy via tie-lines from
other areas. But eventually, the area that is subject to
the load change should balance it without external
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Ali Mohamed Yousef
ISSN: 1991-8763 265 Issue 7, Volume 6, July 2011
Page 2
support. Otherwise there would be economic
conflicts between the areas. Hence each area requires
a separate load frequency controller to regulate the
tie-line power exchange error so that all the areas in
an interconnected power system can set their set
points differently. Another problem is that the
interconnection of the power systems results in huge
increases in both the order of the system and the
number of the tuning controller parameters. As a
result, when modeling such complex high-order
power systems, the model and parameter
approximations cannot be avoided [11-14].
Therefore, the requirement of the LFC is to be robust
against the uncertainties of the system model and the
variations of system parameters in reality. Model
Predictive Control (MPC) refers to a class of
computer control algorithms that utilize an explicit
process model to predict the future response of a
plant. At each control interval an MPC algorithm
attempts to optimize future plant behavior by
computing a sequence of future manipulated variable
adjustments. The first input in the optimal sequence
is then sent into the plant, and the entire calculation
is repeated at subsequent control intervals. Originally
developed to meet the specialized control needs of
power plants and petroleum refineries, MPC
technology can now be found in a wide variety of
application areas including chemicals, food
processing, automotive, and aerospace applications.
Several recent publications provide a good
introduction to theoretical and practical issues
associated with MPC technology [5]. A more
comprehensive overview of nonlinear MPC and
moving horizon estimation, including a summary of
recent theoretical developments and numerical
solution techniques are presented [6].
Model predictive control is also called
recede horizon control [ 8]. The receding horizon
concept is used because at each sampling instant the
optimized control values for the model system over
the prediction horizon are brought up to date, and at
each sampling instant only the first control signal of
the seguence calculated will be used to control the
real system [9,10]. There are two important
parameters in MPC which are prediction horizon and
control horizon. Prediction horizon is the length of
time for the process outputs to approach steady state
values. Also, the control horizon is the number of
discrete time control actions to be optimized along a
future prediction horizon.
The Model Predictive and Fuzzy Logic Control
are applied in the two-area load frequency power
system model. Moreover, comparison between all
controllers at different condition are evaluated. In
general, the engineering tool MATLAB/Simulink is
used to simulate both model predictive and fuzzy
logic control in the power system under study [7].
2 Dynamic Model of the Power
System
Figure 1 shows a block diagram of the ith
area of an N-area power system. Because of
small changes in the load are expected during
normal operation, a linearized area model can be
used for the load-frequency control [15]. The
following one area equivalent model for the
system is modeled. The system investigated
comprises an interconnection of two areas load
frequency control.
Fig. 1: Block diagram of the ith area
The differential equation for the speed governor is
such:
)(1
)(1
)(1
)(1
)(
..
tET
tpT
tfRT
txT
tx i
gi
ci
gi
i
igi
vi
gi
vi
(1)
The differential equation for the turbine generator is
such:
)(1
)(1
)(.
txT
tpT
tp vi
ti
gi
ti
gi (2)
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Ali Mohamed Yousef
ISSN: 1991-8763 266 Issue 7, Volume 6, July 2011
Page 3
The differential equation for the power system is
such:
))()()((
*2
)(2
)(
,
.
tptptp
H
ftf
H
fDtf
digiitie
i
o
i
i
oi
i
(3)
The tie-line power equation is such:
))()(()(1
,
.
tftfTtp ji
N
i
ijitie
(4)
And
)()()( , tfbKtpKtE iiiitieii (5)
Where;
∆𝑓𝑖 =
the incremental frequency deviation for
the ith area;
∆𝑝𝑐𝑖 = the incremental change in speed
changer position for the
ith area;
∆𝑝𝑑𝑖 = the incremental change in load demand
for the ith area;
∆𝑝𝑔𝑖 = the incremental change in power
generation level for the ith area;
∆𝑝𝑡𝑖𝑒 = the incremental tie-line power;
∆𝑥𝑣𝑖 = incremental change in valve position for
the ith area;
∆𝐸𝑖 = the incremental change in the integral
control for the ith area;
𝑓𝑜= the nominal frequency of the system;
𝐷𝑖 = the load frequency constant for the ith area;
𝐻𝑖 = the inertia constant for the ith area;
𝑏𝑖 = the bias constant for the ith area;
𝐾𝑖 =the gain constant for the ith area;
𝑅𝑖 = the regulation constant for the ith area;
𝑇𝑔𝑖= the governor time constant for the ith area;
𝑇𝑖𝑗 = the synchronizing constant between the ith
and jth area;
𝑇𝑡𝑖= the turbine time constant for the ith area;
Let
)()()( 2,1, tptptp tietietie (6)
The overall state vector for two-area load frequency
control system is defined such:
)()(1 tptx tie ; )()( 12 tftx ;
)()( 13 tptx g ;
)()( 14 txtx v )()( 15 tEtx ; )()( 26 tftx ;
)()( 27 tptx g ; )()( 28 txtx v )()( 29 tEtx
The control vector is such:
)(
)(
)(
)()(
2
1
2
1
tp
tp
tu
tutu
c
c
The two-area power system can be written in state-
space form as follows
)()()()(.
tdtButAxtx (7)
Where;
0000000
110
100000
011
000000
0022
00002
0000000
000011
01
0
0000011
00
000000222
0000000
222
2222
22
22
2
2
111
1111
11
11
1
1
1212
bKK
TTRT
TT
H
f
H
Df
H
f
bKK
TTRT
TT
H
f
H
Df
H
f
TT
A
ggg
tt
ooo
ggg
tt
ooo
(8)
9
8
7
6
5
4
3
2
1
x
x
x
x
x
x
x
x
x
x
,
00
10
00
00
00
01
00
00
00
2
1
g
g
T
T
B
,
(9)
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Ali Mohamed Yousef
ISSN: 1991-8763 267 Issue 7, Volume 6, July 2011
Page 4
0
0
0
)(2
0
0
0
)(2
0
)(
2
2
1
1
tpH
f
tpH
f
td
do
do
3 Design of Fuzzy Logic Load
Frequency Control System
Fuzzy interference system (FIS ) consists of
input block, output block and their respective
membership functions. The rules are framed
according to the requirement of the frequency
deviation. More number of rules gives more accurate
results. A normalized values of two inputs
frequency error (deviation) e and change in
frequency error (deviation) ce and defuzzified value
of control command (u) as an output are
considered. Basically, Fuzzy system includes three
processes: a) Normalization b) Fuzzification and c)
Defuzzification. Fig. 2 depicts the stages of fuzzy
system. A centroid method is implemented for
defuzzifization stage. Fuzzification mamdani method
is used.
Fuzzy logic has an advantage over other
control methods due to the fact that it does not
sensitive to plant parameter variations. The fuzzy
logic control approach consists of three stages
,namely fuzzification, fuzzy control rules engine, and
defuzzification. To design the fuzzy logic load
frequency control, the input signals is the frequency
deviation e(k) at sampling time and its change ce(k).
While, its output signal is the change of control
signal U(k) . When the value of the control signal
(U(k-1)) is added to the output signal of fuzzy logic
controller, the result control signal U(k) is obtained.
While the fuzzy membership function variable
signals e , ce, and u are shown in Fig. 3. Fuzzy
control rules are illustrated in table 1. The
membership function shapes of error and error
change are chosen to be identical with triangular
function for fuzzy logic control.
Fig. 2 : The three stages of fuzzy system
Fig. 3: The features of output membership function
Table 1: Fuzzy logic control rules of u .
e
Ce
LN MN SN Z SP MP LP
LN LP LP LP MP MP SP Z
MN LP MP MP MP SP Z SN
SN LP MP SP SP Z SN MN
Z MP MP SP Z SN MN MN
SP MP SP Z SN SN MN
LN SP Z SN MN MN MN LN
LP Z SN MN MN LN LN LN
Where; LN: large negative membership function;
MN: medium negative; SN: small negative; Z: zero;
SP: small positive; MP: medium positive; LP: large
positive.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Ali Mohamed Yousef
ISSN: 1991-8763 268 Issue 7, Volume 6, July 2011
Page 5
4 Model Predictive Control
MPC is a generic term for computer control
algorithms that utilizes an explicit process model to
predict future response of the plant [16 ]. An optimal
input is computed by solving an open-loop optimal
control problem over a finite time horizon, i.e. for a
finite number of future samples. The number of
samples one looks ahead is called the prediction
horizon Np. In some MPC formulations a difference
is made between prediction horizon and control
horizon Nu. The control horizon is then the number
of samples that the optimal input is calculated for.
With a shorter control horizon than prediction
horizon the complexity of the problem can be
reduced. From the calculated input signal only the
first element is applied to the system. This is done at
every time step. The idea is thus to go one step at a
time and check further and further ahead. The
method can be described as ‖repeated open-loop
optimal control in feedback fashion‖.
In an MPC-algorithm there are four important
elements:
4.1 Model prediction
The MPC plant model is defined in discrete time
state space as follows :
)()()()1( kuBkuBkxAkx MDMD
)()( kxCky .
(10)
where )(kx is the state vector, )(ku the input vector,
)(kuMD is called the vector of measured
disturbances, i.e. input signals that are not calculated
by the controller, and )(ky is the output vector.
4.2 Cost function
The cost function is designed depending on what
to minimize. Common is a quadratic cost function
which penalizes both deviation from a state reference
and changes in the control signal and is defined in
the following equation [8,17].
1
0
,,
,,
min p
pppp
N
iiu
T
i
irefix
T
irefi
NrefN
T
NrefN
uQu
xxQxx
xxSxx
(11)
Where
)1()( ikuikuui 𝑥𝑟𝑒𝑓 = state reference
= state predicted
xQ = weight matrices
uQ
=weight matrices
S =weight matrices
A penalty on iu punishes rapid changes in
the input signal, which can be used to reduce
oscillations. According to a penalty on rapid changes
in the input signal, it introduces integral action.
However, this is coupled to the prediction horizon.
Stationary errors can appear even if integral action is
introduced via a penalty on iu if the prediction
horizon is not long enough.
In the cost function stated above, the prediction
horizon (Np) and the control horizon (Nu) are the
same. Instead of a shorter control horizon, there is a
penalty onpNx , which plays a similar role. MPC-
toolbox, which will be used for implementation in
Simulink, uses the formulation where Nu are distinct
from Np. The matrices S, xQ and uQ are weight
matrices who decide the penalty on each term in the
cost function. Most effort is put on minimizing the
term with largest penalty. The general form of the
cost function is defined by Eqn. (12)
(12)
Where;
According to this definition of the cost function,
a simple criterion function will be
(13)
Where ky'
is the predicted output at sampling
time k ,
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Ali Mohamed Yousef
ISSN: 1991-8763 269 Issue 7, Volume 6, July 2011
Page 6
wk is the reference trajectory at sampling
time k and
Now the controller output sequence uopt over the
prediction horizon is obtained by minimization of J
at each sampling instant.
4.3 Constraints
Quadratic Optimization approach (QP) is used to
solve the MPC problem. The QP is a convex
problem, i.e. if a solution is found uniqueness is
guaranteed. To get a QP the constraints need to be
linear [ 17]. They are thus on the form:
givenxo
1,.....,0,maxmin pi Niuuu
pi NiyxWy ,...,1max,min
(14)
where the matrix W is the output states.
4.4 Optimization problem and algorithm
The optimization vector is:
Tp
TT NkukuU )1(),...,( .
If a shorter control horizon than prediction horizon is
used, it is assumed that
)1()( uNkuiku for all uNi . The
problem now needs to be rewritten in terms of U
only. This is in principle straightforward since [ 16]:
1
0
1
))(
)(()0()(
k
j MDMD
jk
k
juB
juBAxAkx
(15)
Finally the optimization problem becomes:
UhHUUU TT min (16)
where the matrices H and the vector h are build up
by xref , uref , x0,Qx,Qu, S, A, B and C.
The MPC-algorithm can now be summarized as:
a. Measure the current state )(kx or estimate it
using an observer.
b. Solve the k-th optimization problem to obtain
Tp
TT NkukuU )1(),...,( .
c. Apply )(ku to the system.
d. Update time k=k+1 and repeat from step 1.
Figure 4 depicts the proposed MPC applied on the
two-area load frequency control model.
Fig. 4: The proposed MPC of the two-area load
frequency control.
5 Digital Simulation Results
The block diagram of the two-area load
frequency control with the proposed MPC is shown
in Fig. 4. The entire system has been simulated and
subjected to different parameters changes on the
digital computer using the Matlab program and
Simulink software package. The power system
frequency deviations are obtained. A comparison
between the power system responses using the
conventional FLC and the proposed MPC are
evaluated. The system investigated parameters are
[1]:
fo=60 HZ R1=R2=2.4 HZ/per unit MW
Tg1=Tg2=0.08 s Tr=10.0s Tt1=Tt2 =0.3s
TR=5 s D1=D2=0.00833 Mw/HZ
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Ali Mohamed Yousef
ISSN: 1991-8763 270 Issue 7, Volume 6, July 2011
Page 7
T1=48.7s T2=0.513s , Tp1= Tp2=20
Kp1=120; a12=-1;
Kp2=120; T12=0.545 MW
From Eqns. (8, 9), the A matrix and B input vector
are calculated as:
And , choice of MPC :
control horizon = 18
prediction horizon = 5
Figure 5 shows the system responses due to 5% load
disturbance in area-1 without any control. Fig. 6
displays the frequency deviation response in pu. of
area-1 due to 0.05p.u. load disturbance in area-1 of
the two- area power system with FLC and proposed
MPC. Fig.7 shows the frequency deviation response
in pu. of area-2 due to 0.05p.u.load disturbance in
area-1 of the two- area power system with FLC and
proposed MPC. Fig. 8 shows the tie-line power
deviation response in pu. of area-1 due to
0.05p.u.load disturbance in area-1 of the two- area
power system with FLC and proposed MPC. Fig. 9
shows the frequency deviation response in pu. of
area -1 due to 0.05p.u.load disturbance in area-2 of
the two- area power system. Also, Fig. 10 depicts the
frequency deviation response in pu. of area -2 due
to 0.05p.u.load disturbance in area-2 of the two- area
power system. Fig.11 depicts the tie-line power
deviation response in pu. due to 0.05p.u.load
disturbance in area-2 of the two- area power system
with FLC and proposed MPC. Table 2 discribes the
settling time and under shoot calculation with FLC
and MPC.
0 10 20 30 40 50 60 70 80-0.18
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
Time in Sec.
System responses without controller
F1 dev.
F2 dev.
P-tie dev.
Fig. 5: The system responses in pu. due to 5%
disturbance without any control
Fig. 6: Frequency deviation response in pu. of area-1 due
to 0.05 p.u. load disturbance in area-1 of the two- area
power system with FLC and proposed MPC.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Ali Mohamed Yousef
ISSN: 1991-8763 271 Issue 7, Volume 6, July 2011
Page 8
Fig. 7 : Frequency deviation response in pu. of area-
2 due to 0.05 p.u. load disturbance in area-1 of the
two- area power system with FLC and proposed
MPC
Fig. 8:Tie-line power deviation response in pu. due
to 0.05p.u.load disturbance in area-1 of the two- area
power system with FLC and proposed MPC .
Fig. 9: : Frequency deviation response in pu. of
area-1 due to 0.05 p.u. load disturbance in area-2 of
the two- area power system with FLC and proposed
MPC
Fig. 10: Frequency deviation response in pu. of
area-2 due to 0.05 p.u. load disturbance in area-2 of
the two- area power system with FLC and proposed
MPC.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Ali Mohamed Yousef
ISSN: 1991-8763 272 Issue 7, Volume 6, July 2011
Page 9
Fig. 11: Tie-line power deviation response in pu.
due to 0.05p.u.load disturbance in area-2 of the two-
area power system with FLC and proposed MPC.
Table 2: The settling time and under shoot calculation with FLC and MPC. 5% Load disturbance in area No:1 5% Load disturbance in area No:2
FLC MPC FLC MPC
(Sec.)
Under Shoot
in pu.
(Sec.)
Under
Shoot in
pu.
(Sec.)
Under
Shoot in pu.
(Sec.)
Under
Shoot in
pu.
40 -0.12 10 -0.07 40 -0.007 10 -0.001
41 -0.13 20 -0.1 42 -0.008 10 -0.002
25 -0.025 20 -0.01 42 -0.008 10 -0.002
Where; Ts =The settling time in Sec.
6 Discussions
The Fuzzy Interference System (FIS) matrix
for fuzzy logic controoler is devolped, considering
49 rules as in table-1 by using Gaussian, Trapizoidal
and Triangular membership functions. Moreover, a
MPC simulink is designed based on power system
model , control horizon and prediction horizon.
Various transient response curves of 1f , 2f ,
linetieP are drawn and comparative studies have
been made. The following points may be noted:
1. From Fig. 5 notice that the two-area load
frequency control power system has steady
state error without any control.
2. From figures 6:11 and table 2, the frequency
deviation responses based on proposed
MPC is better than fuzzy logic control in
terms of fast response and small settling
time.
3. The tie line power is also fastly decreased in
case of MPC than fuzzy logic control.
4. The performance of the MPC is seen in
figures 6:11 and table 2 was effective
enough to eliminate the oscillation after 10
Sec.
5. The performance of the FLC is seen in
figures 6:11 and table 2 was not effective
enough to eliminate the oscillation after 40
Sec.
6. In order to have a better prediction of the
future behavior of the plant, the prediction
horizon should be more than the period of
the system.
7. Model predictive control has been shown to
be successful in addressing many large scale
non-linear control problems.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Ali Mohamed Yousef
ISSN: 1991-8763 273 Issue 7, Volume 6, July 2011
Page 10
7 Conclusions
This paper addressed the load frequency
control problem of interconnected power systems.
Two control schemes are proposed for the system.
The design of the proposed control schemes was
based on fuzzy logic and model predictive controls.
The load-frequency control system based MPC for
enhancing power system dynamic performances after
applying several disturbances was evaluated. The
proposed controllers are robust and gives good
transient as well as steady -state performance. To
validate the effectiveness of the proposed controller a
comparison among the FLC and proposed MPC
controller is obtained. The proposed controller
proves that it is robust to variations in disturbance
changes from area-1 and area-2. The digital
simulation results proved that the effectiveness of the
proposed MPC over the FLC through a wide range of
load disturbances. The superiority of the proposed
MPC is embedded in sense of fast response with less
overshoot and / or undershoot and less settling time.
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]17].
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Ali Mohamed Yousef
ISSN: 1991-8763 274 Issue 7, Volume 6, July 2011
Page 11
power system‖, Master thesis of Engineering,
Faculty of Built Environment and
Engineering, Queensland University, 2007
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Ali Mohamed Yousef
ISSN: 1991-8763 275 Issue 7, Volume 6, July 2011